Average Error: 0.5 → 0.4
Time: 14.6s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot \left(\sqrt{\frac{1}{k}} \cdot {\left(\sqrt{2 \cdot \left(n \cdot \pi\right)}\right)}^{\left(-k\right)}\right) \]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot \left(\sqrt{\frac{1}{k}} \cdot {\left(\sqrt{2 \cdot \left(n \cdot \pi\right)}\right)}^{\left(-k\right)}\right)

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (*
  (sqrt (* n (* 2.0 PI)))
  (* (sqrt (/ 1.0 k)) (pow (sqrt (* 2.0 (* n PI))) (- k)))))
double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}
↓
double code(double k, double n) {
	return sqrt(n * (2.0 * ((double) M_PI))) * (sqrt(1.0 / k) * pow(sqrt(2.0 * (n * ((double) M_PI))), -k));
}

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

  3. Taylor expanded in n around 0 3.5

    \[\leadsto \color{blue}{e^{0.5 \cdot \left(\left(1 - k\right) \cdot \left(\log n + \log \left(2 \cdot \pi\right)\right)\right)} \cdot \sqrt{\frac{1}{k}}} \]

  4. Simplified0.5

    \[\leadsto \color{blue}{{\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{\left(1 - k\right)} \cdot \sqrt{\frac{1}{k}}} \]
  5. Using strategy rm

  6. Applied sub-neg_binary640.5

    \[\leadsto {\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{\color{blue}{\left(1 + \left(-k\right)\right)}} \cdot \sqrt{\frac{1}{k}} \]

  7. Applied unpow-prod-up_binary640.4

    \[\leadsto \color{blue}{\left({\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{1} \cdot {\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{\left(-k\right)}\right)} \cdot \sqrt{\frac{1}{k}} \]

  8. Applied associate-*l*_binary640.4

    \[\leadsto \color{blue}{{\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{1} \cdot \left({\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{\left(-k\right)} \cdot \sqrt{\frac{1}{k}}\right)} \]

  9. Simplified0.4

    \[\leadsto {\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{1} \cdot \color{blue}{\left(\sqrt{\frac{1}{k}} \cdot {\left(\sqrt{2 \cdot \left(n \cdot \pi\right)}\right)}^{\left(-k\right)}\right)} \]

  10. Final simplification0.4

    \[\leadsto \sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot \left(\sqrt{\frac{1}{k}} \cdot {\left(\sqrt{2 \cdot \left(n \cdot \pi\right)}\right)}^{\left(-k\right)}\right) \]

Reproduce

herbie shell --seed 2021206 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))